Integrand size = 23, antiderivative size = 31 \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx=\frac {\sqrt {1+x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{\sqrt {-1-x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {432, 430} \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx=\frac {\sqrt {x^2+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {x}{\sqrt {2}}\right ),-2\right )}{\sqrt {-x^2-1}} \]
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Rule 430
Rule 432
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+x^2} \int \frac {1}{\sqrt {2-x^2} \sqrt {1+x^2}} \, dx}{\sqrt {-1-x^2}} \\ & = \frac {\sqrt {1+x^2} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{\sqrt {-1-x^2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx=-\frac {i \sqrt {1+x^2} \operatorname {EllipticF}\left (i \text {arcsinh}(x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1-x^2}} \]
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Time = 2.45 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {i F\left (i x , \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-x^{2}-1}}{2 \sqrt {x^{2}+1}}\) | \(34\) |
elliptic | \(-\frac {i \sqrt {\left (x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {x^{2}+1}\, \sqrt {-2 x^{2}+4}\, F\left (i x , \frac {i \sqrt {2}}{2}\right )}{2 \sqrt {-x^{2}-1}\, \sqrt {-x^{2}+2}\, \sqrt {x^{4}-x^{2}-2}}\) | \(74\) |
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none
Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {-2} F(\arcsin \left (\frac {1}{2} \, \sqrt {2} x\right )\,|\,-2) \]
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\[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx=\int \frac {1}{\sqrt {2 - x^{2}} \sqrt {- x^{2} - 1}}\, dx \]
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\[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 2} \sqrt {-x^{2} - 1}} \,d x } \]
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\[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx=\int { \frac {1}{\sqrt {-x^{2} + 2} \sqrt {-x^{2} - 1}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx=\int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-x^2}} \,d x \]
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